Contour expression explanation

Question

$$ \begin{align} & \int_\gamma zdz \end{align} $$ $$\\ \gamma = [e,1]+[1,-1+\sqrt3] $$

contour $\gamma$ is defined as above and I can't understand it. Could someone please explain it to me and provide me with a $\gamma(t)$=... form of the contour.

Answer 1

(Answering the question in your comment)

The chain $\gamma:=\sigma(e,1)+\sigma(1,-1+\sqrt{3})$ is a path (not a "contour") beginning at $a(\gamma)=e$ and ending at $b(\gamma)=-1+\sqrt{3}$.

On the other hand, the function $f(z):=z$ to be integrated along $\gamma$ has a primitive $F(z):={z^2\over2}$, valid in all of ${\mathbb C}$. It follows by one of the first principles of path-integration that $$\int_\gamma z\>dz=\int_\gamma f(z)\>dz=F\bigl(b(\gamma)\bigr)-F\bigl(a(\gamma)\bigr)={z^2\over2}\Biggr|_e^{-1+\sqrt{3}}=\ldots\quad.$$

Answer 2

Presumably a path that goes from $e$ to $1$ and then from $1$ to $\sqrt{3}-1$.

You should be able to parametrize these two parts individually...